[tex]\log _a(b)=c[/tex]
Using the definition of the base of a log:
[tex]\log _a(b)=c\equiv a^c=b[/tex]
According to this, c could be a negative number, here's why:
Let's try to find the following logarithm:
[tex]\log _2(\frac{1}{8})=c[/tex]
Using the definition of the base of a log:
[tex]2^c=\frac{1}{8}[/tex]
We can express 8 as 2³:
[tex]2^c=\frac{1}{2^3}[/tex]
Using the following property:
[tex]a^{-x}=\frac{1}{a^x}[/tex]
We can express the equation as:
[tex]2^c=2^{-3}[/tex]
From this, we can conclude that c = -3, therefore it is possible for x to equal a negative number.
